Biases in Moments of Satake Parameters and in Zeros near the Central - - PowerPoint PPT Presentation

Bias: ECs Central Point Refs Bias: New Families

Biases in Moments of Satake Parameters and in Zeros near the Central Point in Families of L-Functions

Steven J. Miller (Williams College)

sjm1@williams.edu, Steven.Miller.MC.96@aya.yale.edu

http://web.williams.edu/Mathematics/sjmiller/public_html/

Lightening Talk: Computational Aspects of L-functions ICERM, Providence, RI, November 10, 2015

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Bias: ECs Central Point Refs Bias: New Families

Bias Conjecture for Elliptic Curves With Blake Mackall (Williams), Christina Rapti (Bard) and Karl Winsor (Michigan)

Emails: Blake.R.Mackall@williams.edu, cr9060@bard.edu, krlwnsr@umich.edu.

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Last Summer: Families and Moments

A one-parameter family of elliptic curves is given by E : y 2 = x3 + A(T)x + B(T) where A(T), B(T) are polynomials in Z[T]. Each specialization of T to an integer t gives an elliptic curve E(t) over Q. The r th moment of the Fourier coefficients is Ar,E(p) =

• t

mod p

aE(t)(p)r.

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Negative Bias in the First Moment

A1,E(p) and Family Rank (Rosen-Silverman) If Tate’s Conjecture holds for E then lim

X→∞

1 X

• p≤X

A1,E(p) log p p = −rank(E/Q). By the Prime Number Theorem, A1,E(p) = −rp + O(1) implies rank(E/Q) = r.

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Bias: ECs Central Point Refs Bias: New Families

Bias Conjecture

Second Moment Asymptotic (Michel) For families E with j(T) non-constant, the second moment is A2,E(p) = p2 + O(p3/2). The lower order terms are of sizes p3/2, p, p1/2, and 1. In every family we have studied, we have observed: Bias Conjecture The largest lower term in the second moment expansion which does not average to 0 is on average negative.

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Bias: ECs Central Point Refs Bias: New Families

Preliminary Evidence and Patterns

Let n3,2,p equal the number of cube roots of 2 modulo p, and set c0(p) =

−3

p

• +

3

p

• p, c1(p) =
• x mod p

x3−x

p

2 , c3/2(p) = p

x(p)

4x3+1

p

• .

Family A1,E(p) A2,E(p) y 2 = x3 + Sx + T p3 − p2 y 2 = x3 + 24(−3)3(9T + 1)2

• 2p2−2p

p≡2 mod 3 p≡1 mod 3

y 2 = x3 ± 4(4T + 2)x

• 2p2−2p

p≡1 mod 4 p≡3 mod 4

y 2 = x3 + (T + 1)x2 + Tx p2 − 2p − 1 y 2 = x3 + x2 + 2T + 1 p2 − 2p − −3

p

• y 2 = x3 + Tx2 + 1

−p p2 − n3,2,pp − 1 + c3/2(p) y 2 = x3 − T 2x + T 2 −2p p2 − p − c1(p) − c0(p) y 2 = x3 − T 2x + T 4 −2p p2 − p − c1(p) − c0(p) y 2 = x3 + Tx2 − (T + 3)x + 1 −2cp,1;4p p2 − 4cp,1;6p − 1

where cp,a;m = 1 if p ≡ a mod m and otherwise is 0.

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Lower order terms and average rank

1 N

2N

• t=N
• γt

φ

• γt

log R 2π

• =

φ(0) + φ(0) − 2 N

2N

• t=N
• p

log p log R 1 p

• φ

log p log R

• at(p)

− 2 N

2N

• t=N
• p

log p log R 1 p2 φ 2 log p log R

• at(p)2 + O

log log R log R

• .

φ(x) ≥ 0 gives upper bound average rank. Expect big-Oh term Ω(1/ log R).

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Bias: ECs Central Point Refs Bias: New Families

Implications for Excess Rank

Katz-Sarnak’s one-level density statistic is used to measure the average rank of curves over a family. More curves with rank than expected have been

• bserved, though this excess average rank vanishes

in the limit. Lower-order biases in the moments of families explain a small fraction of this excess rank phenomenon.

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Bias: ECs Central Point Refs Bias: New Families

Methods for Obtaining Explicit Formulas

For a family E : y 2 = x3 + A(T)x + B(T), we can write aE(t)(p) = −

• x

mod p

x3 + A(t)x + B(t) p

• where
• ·

p

• is the Legendre symbol modp given by

x p

• =

     1 if x is a non-zero square modulo p if x ≡ 0 mod p −1

• therwise.

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Lemmas on Legendre Symbols

Linear and Quadratic Legendre Sums

• x

mod p

ax + b p

• = 0

if p ∤ a

• x

mod p

ax2 + bx + c p

• =

   −

• a

p

• if p ∤ b2 − 4ac

(p − 1)

• a

p

• if p | b2 − 4ac

Average Values of Legendre Symbols The value of

• x

p

• for x ∈ Z, when averaged over all

primes p, is 1 if x is a non-zero square, and 0 otherwise.

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Bias: ECs Central Point Refs Bias: New Families

Rank 0 Families

Theorem (MMRW’14): Rank 0 Families Obeying the Bias Conjecture For families of the form E : y 2 = x3 + ax2 + bx + cT + d, A2,E(p) = p2 − p

• 1 +

−3 p

• +

a2 − 3b p

• .

The average bias in the size p term is −2 or −1, according to whether a2 − 3b ∈ Z is a non-zero square.

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Bias: ECs Central Point Refs Bias: New Families

Families with Rank

Theorem (MMRW’14): Families with Rank For families of the form E : y 2 = x3 + aT 2x + bT 2,

A2,E(p) = p2 − p

• 1 +
• −3

p

• +
• −3a

p

• −
• x(p)
• x3+ax

p

2 .

These include families of rank 0, 1, and 2. The average bias in the size p terms is −3 or −2, according to whether −3a ∈ Z is a non-zero square.

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Bias: ECs Central Point Refs Bias: New Families

Families with Rank

Theorem (MMRW’14): Families with Complex Multiplication For families of the form E : y 2 = x3 + (aT + b)x, A2,E(p) = (p2 − p)

• 1 +

−1 p

• .

The average bias in the size p term is −1. The size p2 term is not constant, but is on average p2, and an analogous Bias Conjecture holds.

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Bias: ECs Central Point Refs Bias: New Families

Families with Unusual Distributions of Signs

Theorem (MMRW’14): Families with Unusual Signs For the family E : y 2 = x3 + Tx2 − (T + 3)x + 1, A2,E(p) = p2 − p

• 2 + 2

−3 p

• − 1.

The average bias in the size p term is −2. The family has an usual distribution of signs in the functional equations of the corresponding L-functions.

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Bias: ECs Central Point Refs Bias: New Families

The Size p3/2 Term

Theorem (MMRW’14): Families with a Large Error For families of the form E : y 2 = x3 + (T + a)x2 + (bT + b2 − ab + c)x − bc, A2,E(p) = p2 − 3p − 1 + p

• x

mod p

−cx(x + b)(bx − c) p

• The size p3/2 term is given by an elliptic curve

coefficient and is thus on average 0. The average bias in the size p term is −3.

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Bias: ECs Central Point Refs Bias: New Families

General Structure of the Lower Order Terms

The lower order terms appear to always have no size p3/2 term or a size p3/2 term that is on average 0; exhibit their negative bias in the size p term; be determined by polynomials in p, elliptic curve coefficients, and congruence classes of p (i.e., values

• f Legendre symbols).

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Bias: ECs Central Point Refs Bias: New Families

New Families: Work in Progress

Dirichlet characters of prime level: bias +1. Holomorphic cusp forms: bias −1/2. r th Symmetric Power Fr,X,δ,q: bias +1/48. (With Megumi Asada and Eva Fourakis (Williams), Kevin Yang (Harvard).)

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Bias: ECs Central Point Refs Bias: New Families

Finite Conductor Models at Central Point With Owen Barrett and Blaine Talbut (Chicago), Gwyn Moreland (Michigan), Nathan Ryan (Bucknell)

Emails: owen.barrett@yale.edu, gwynm@umich.edu, blainetalbut@gmail.com, nathan.ryan@bucknell.edu. Excised Orthogonal Ensemble joint with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith. Numerical experiments

• ngoing with Nathan Ryan.

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RMT: Theoretical Results (N → ∞)

0.5 1 1.5 2 0.5 1 1.5 2

1st normalized evalue above 1: SO(even)

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Bias: ECs Central Point Refs Bias: New Families

RMT: Theoretical Results (N → ∞)

0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2

1st normalized evalue above 1: SO(odd)

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Rank 0 Curves: 1st Norm Zero: 14 One-Param of Rank 0

1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2

Figure 4a: 209 rank 0 curves from 14 rank 0 families, log(cond) ∈ [3.26, 9.98], median = 1.35, mean = 1.36

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Rank 0 Curves: 1st Norm Zero: 14 One-Param of Rank 0

0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4

Figure 4b: 996 rank 0 curves from 14 rank 0 families, log(cond) ∈ [15.00, 16.00], median = .81, mean = .86.

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Bias: ECs Central Point Refs Bias: New Families

Spacings b/w Norm Zeros: Rank 0 One-Param Families over Q(T)

All curves have log(cond) ∈ [15, 16]; zj = imaginary part of jth normalized zero above the central point; 863 rank 0 curves from the 14 one-param families of rank 0 over Q(T); 701 rank 2 curves from the 21 one-param families of rank 0 over Q(T). 863 Rank 0 Curves 701 Rank 2 Curves t-Statistic Median z2 − z1 1.28 1.30 Mean z2 − z1 1.30 1.34

• 1.60

StDev z2 − z1 0.49 0.51 Median z3 − z2 1.22 1.19 Mean z3 − z2 1.24 1.22 0.80 StDev z3 − z2 0.52 0.47 Median z3 − z1 2.54 2.56 Mean z3 − z1 2.55 2.56

• 0.38

StDev z3 − z1 0.52 0.52

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Bias: ECs Central Point Refs Bias: New Families

Spacings b/w Norm Zeros: Rank 2 one-param families over Q(T)

All curves have log(cond) ∈ [15, 16]; zj = imaginary part of the jth norm zero above the central point; 64 rank 2 curves from the 21 one-param families of rank 2 over Q(T); 23 rank 4 curves from the 21 one-param families of rank 2 over Q(T). 64 Rank 2 Curves 23 Rank 4 Curves t-Statistic Median z2 − z1 1.26 1.27 Mean z2 − z1 1.36 1.29 0.59 StDev z2 − z1 0.50 0.42 Median z3 − z2 1.22 1.08 Mean z3 − z2 1.29 1.14 1.35 StDev z3 − z2 0.49 0.35 Median z3 − z1 2.66 2.46 Mean z3 − z1 2.65 2.43 2.05 StDev z3 − z1 0.44 0.42

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Bias: ECs Central Point Refs Bias: New Families

Rank 2 Curves from Rank 0 & Rank 2 Families over Q(T)

All curves have log(cond) ∈ [15, 16]; zj = imaginary part of the jth norm zero above the central point; 701 rank 2 curves from the 21 one-param families of rank 0 over Q(T); 64 rank 2 curves from the 21 one-param families of rank 2 over Q(T). 701 Rank 2 Curves 64 Rank 2 Curves t-Statistic Median z2 − z1 1.30 1.26 Mean z2 − z1 1.34 1.36 0.69 StDev z2 − z1 0.51 0.50 Median z3 − z2 1.19 1.22 Mean z3 − z2 1.22 1.29 1.39 StDev z3 − z2 0.47 0.49 Median z3 − z1 2.56 2.66 Mean z3 − z1 2.56 2.65 1.93 StDev z3 − z1 0.52 0.44

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Bias: ECs Central Point Refs Bias: New Families

New Model for Finite Conductors

Replace conductor N with Neffective.

⋄ Arithmetic info, predict with L-function Ratios Conj. ⋄ Do the number theory computation.

Excised Orthogonal Ensembles.

⋄ L(1/2, E) discretized. ⋄ Study matrices in SO(2Neff ) with |ΛA(1)| ≥ ceN.

Painlevé VI differential equation solver.

⋄ Use explicit formulas for densities of Jacobi ensembles. ⋄ Key input: Selberg-Aomoto integral for initial conditions.

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Bias: ECs Central Point Refs Bias: New Families

Modeling lowest zero of LE11(s, χd) with 0 < d < 400, 000

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 1 1.5 2

Lowest zero for LE11(s, χd) (bar chart), lowest eigenvalue

• f SO(2N) with Neff (solid), standard N0 (dashed).

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Bias: ECs Central Point Refs Bias: New Families

Modeling lowest zero of LE11(s, χd) with 0 < d < 400, 000

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 1 1.5 2

Lowest zero for LE11(s, χd) (bar chart); lowest eigenvalue

• f SO(2N): Neff = 2 (solid) with discretisation, and

Neff = 2.32 (dashed) without discretisation.

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Bias: ECs Central Point Refs Bias: New Families

Effective Matrix Size: Families with Unitary Symplectic Monodromy

L-function attached to quadratic Dirichlet character.

⋄ L(χ, s) =

p<∞ (1 − χ(p)p−s)−1.

L-function attached to symmetric power.

⋄ L(Symrf, s) =

p<∞ Lp(Symrf, s).

Compute 1-level Density: Study distribution of zeros

⋄ D1,ϕ(F) = #F−1 ·

f∈F

• ρf =1/2+iγf ϕ(γf · log Q

2π )

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Bias: ECs Central Point Refs Bias: New Families

Integral Representation of One-Level Density

We bound conductors of families by a parameter X ⋄ For quadratic Dirichlet characters, we have: Theorem

The One-Level Density is represented by the integral kernel K(τ) = 1 − sin(2πτ) 2πτ + 1 − cos(2πτ) Λ−1 log X + O

• 1

log2 X

• for Λ < 0.

Similarly for the family of quadratic twists of Symrf.

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Deducing Effective Matrix Size

Matching with integral kernel of matrix groups. ⋄ π

N · K1,USp(2N)(t) = 1 − sin(2πt) 2πt

+ 1−cos(2πt)

2N

+ . . . ⋄ π

N · K1,SO(2N+1)(t), same leading term.

Note π N · (K1,SO(2N+1) − K1,USp(2(−N))) ∼ 1 − cos(2πt) 2N Unitary Symplectic Families behave like SO(2N + 1) for bounded X. Similarly for quadratic twists of Sym2f.

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Bias: ECs Central Point Refs Bias: New Families

Excised Orthogonal Ensemble

As before, let F be those quadratic twists of L(E, s). Idea: interpret L(E, 1

2 + it) as an integral kernel.

Taylor Series expansion: L(E, s) = L(E, 1 2) + L′(E, 1 2)(s − 1 2) + . . . Goal: match power series coefficients with that of chH(eiθ). Amalgamate integral kernels together: attach to F a product distribution

E∈F

∞

0 L(E, 1 2 + it)dt.

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Bias: ECs Central Point Refs Bias: New Families

Excised Orthogonal Ensemble (continued)

We deduce Theorem

Let FX be those quadratic twists of an elliptic curve E/Q of conductor N < X. If supn

• L(n)(E, 1

2) − ch(n)(1)

• < δ, then
• D1,FX − D1,MN(X)
• L2 < ε.

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Bias: ECs Central Point Refs Bias: New Families

References

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Bias: ECs Central Point Refs Bias: New Families

References

Biases:

1- and 2-level densities for families of elliptic curves: evidence for the underlying group symmetries, Compositio Mathematica 140 (2004), 952–992. http://arxiv.org/pdf/math/0310159. Variation in the number of points on elliptic curves and applications to excess rank, C. R. Math. Rep. Acad.

• Sci. Canada 27 (2005), no. 4, 111–120. http://arxiv.org/abs/math/0506461.

Investigations of zeros near the central point of elliptic curve L-functions, Experimental Mathematics 15 (2006), no. 3, 257–279. http://arxiv.org/pdf/math/0508150. Lower order terms in the 1-level density for families of holomorphic cuspidal newforms, Acta Arithmetica 137 (2009), 51–98. http://arxiv.org/pdf/0704.0924v4. Moments of the rank of elliptic curves (with Siman Wong), Canad. J. of Math. 64 (2012), no. 1, 151–182. http://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/ mwMomentsRanksEC812final.pdf 35

Bias: ECs Central Point Refs Bias: New Families

References

Central:

Investigations of zeros near the central point of elliptic curve L-functions, Experimental Mathematics 15 (2006), no. 3, 257–279. http://arxiv.org/pdf/math/0508150.pdf. The lowest eigenvalue of Jacobi Random Matrix Ensembles and Painlevé VI, (with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith), Journal of Physics A: Mathematical and Theoretical 43 (2010) 405204 (27pp). http://arxiv.org/pdf/1005.1298v2. Models for zeros at the central point in families of elliptic curves (with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith), J. Phys. A: Math. Theor. 45 (2012) 115207 (32pp). http://arxiv.org/ pdf/1107.4426.pdf. 36

Bias: ECs Central Point Refs Bias: New Families

Bias Conjecture for Elliptic Curves With Megumi Asada and Eva Fourakis (Williams), Kevin Yang (Harvard)

Emails: maa2@williams.edu, erf1@williams.edu, kevinyang@college.harvard.edu.

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Bias: ECs Central Point Refs Bias: New Families

Summary of Results

Dirichlet characters of prime level: bias +1. Holomorphic cusp forms: bias −1/2. r th Symmetric Power Fr,X,δ,q: bias +1/48.

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Bias: ECs Central Point Refs Bias: New Families

Dirichlet Family Fq

Definition Prime q ∈ Z and Fq = {χ = χ0(q)} is the family of nontrivial Dirichlet characters of conductor q. The second moment at p is M2(Fq; p) :=

• χ∈Fq

χ2(p). Goal: Compute asymptotics for the sum M2,X(Fq) =

• p<X

M2(Fq; p) =

• p<X
• χ∈Fq

χ2(p).

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Results for Fq

Theorem Family Fq has positive bias in the second moment of +1. Have M2(Fq; p) :=

χ∈Fq χ2(p).

From orthogonality relations: M2(Fq; p) =

• q − 2

if p ≡ ±1(q); −1 if p ≡ ±1(q), Thus

• p<X

M2(Fq; p) =

• p<X

p≡±1(q)

(q − 2) −

• p<X

p≡±1(q)

1. Main term size π(X).

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Bias: ECs Central Point Refs Bias: New Families

Cuspidal Newforms

Fix level q = 1. For weight k, consider an orthonormal basis Bk,q(χ0) of Hk,q(χ0), the space of holomorphic cusp forms on the surface Γ0\h of level k and trivial nebentypus. Family FX :=

• k<X

k≡0(2)

Bk,q=1(χ0).

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An Important Tool: Petersson Trace Formula

For any n, m ≥ 1, we have

Γ(k − 1) (4πp)k−1

• f∈Bk,q(χ0)

|λf (p)|2 = δ(p, p)+2πi−k

c≡0(q)

Sc(p, p) c Jk−1 4πp c

• where λf (n) is the n-th Hecke eigenvalue of f,

δ(m, n) is Kronecker’s delta, Sc(m, n) is the classical Kloosterman sum, and Jk−1(t) is the k-Bessel function.

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Cusp Newform: F<X

We gain asymptotic control over Jk−1(t) by averaging over even weights k. M2(FX; p) =

• k∗<X

M2(Hk,1(χ0); p) =

• k∗<X
• f∈Bk,1(χ0)

|λf(p)|2 where

k∗<X denotes summing over even k.

Theorem Let ϕ ∈ C∞

0 (R>0) be real-valued, and let X > 1. Then

4

• k≡0(2)

ϕ k − 1 X

• Jk−1(t) = ϕ

t X

• +

t 6X 3 ϕ(2) t X

• 43

Bias: ECs Central Point Refs Bias: New Families

Cusp Newform: F<X

To handle Sc(m, n), we instead compute M2 (FX; δ) =

• p<X δ

M2 (FX; p) · log p. After several substitutions and iterations of integration by parts, M2(FX; δ) = 1 2X 1+δ − X 1+δ 2 log2 X δ + O X 1+δ log3 X δ

• yields a bias of −1/2.

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Bias: ECs Central Point Refs Bias: New Families

Varying the Level: FX; δ; ǫ

Can also vary the level: M2(FX; δ; ε) =

• q<X ε

M2(Fq,X; δ) =

• q<X ε
• p<X δ
• k∗<X
• f∈Bk,q(χ0)

|λf(p)|2 · log p = 1 2X 1+δ+ε − X 1+δ+ε 2 log2 X δ + O X 1+δ+ε log3 X δ

• .

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Bias: ECs Central Point Refs Bias: New Families

Symmetric Lift Family

Fix a square-free level q and study for δ > 0 Fr,X,δ,q =

• k<X δ

Symr H∗

k,q(χ0)

• .

Second moment: for ε > 0: M2,ε(Fr,X,δ,q) = 1 ϕ(q)

• p<X ε
• k<X δ

 

• f∈H∗

k,q(χ0)

λ2

Symrf(p)

  , find bias of +1/48 in M2,ε(Fr,X,δ) = lim

q→∞ q sq−free

M2,ε(Fr,X,δ,q).

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